⚛️ general relativity

Dynamical Love numbers of analogue rotating black and white holes

This paper calculates the complex-valued dynamical tidal response coefficients for 2+1D analogue black and white holes in rotating bathtub flows, revealing that the black hole's tidal Love number exhibits logarithmic running while its white hole counterpart's response is the complex conjugate of the black hole's.

Original authors: Satadal Datta, Wei-Can Syu, Da-Shin Lee

Published 2026-08-25

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CC BY 4.0

Original authors: Satadal Datta, Wei-Can Syu, Da-Shin Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a massive object, like a planet or a star, sits within the gravitational pull of a neighbor, it does not remain perfectly round. The neighbor's gravity stretches and squeezes it, creating a tidal bulge. In the universe, this stretching is a constant dance between binary stars and orbiting black holes. Scientists measure how much an object deforms under this stress using a value called a tidal Love number. For decades, a stubborn puzzle has existed in the study of black holes. According to the standard theory of gravity, a black hole is so simple and rigid that it should not deform at all; its tidal Love number should be exactly zero. However, recent observations of gravitational waves have hinted that real black holes might behave differently, perhaps holding secrets about the nature of space and time that our current theories cannot yet explain. To investigate this without needing to travel to a distant black hole, physicists have turned to the laboratory, creating miniature models of these cosmic monsters using flowing fluids.

In a new study, researchers have taken this concept of fluid models a step further by calculating how these artificial black holes respond to changing forces over time, rather than just static ones. They focused on a specific setup known as a "draining bathtub," where water spirals inward toward a drain, creating a flow that mimics the way space-time drags around a rotating black hole. In this fluid model, the point where the water flows faster than the speed of sound acts as an event horizon, a boundary from which nothing, not even sound waves, can escape. The researchers also explored the reverse scenario, a "fountaining bathtub" where water flows outward from a source, creating an analogue white hole, a theoretical object that nothing can enter. By analyzing the mathematics of sound waves moving through these rotating flows, the team calculated the dynamical tidal response coefficients, which describe how the fluid system deforms and absorbs energy when subjected to a shifting tidal force.

The results revealed a striking difference between these fluid models and the black holes predicted by Einstein's theory of general relativity. While standard black holes are expected to have a tidal Love number of zero, the analogue black holes in the study were found to have a non-zero response. Specifically, the researchers discovered that the deformation of the fluid black hole depends on the frequency of the tidal force and the rotation speed of the flow. In certain conditions, particularly when the system is static or nearly so, the tidal response does not settle into a single fixed value but instead changes logarithmically as the distance from the center changes. This means the "stiffness" of the analogue black hole is not a simple constant but varies in a specific, predictable way. Furthermore, the study showed that while the analogue black hole absorbs energy from the tidal field, the analogue white hole, which is essentially the time-reversed version of the black hole, reflects that energy in a complementary manner.

The team found that the mathematical description of these responses involves complex numbers, where the real part represents the deformation of the object and the imaginary part represents the dissipation of energy. In the fluid models, the energy dissipation vanished for static cases, but the deformation remained active and significant. This stands in sharp contrast to the behavior of real black holes in Einstein's gravity, where static tidal deformations are predicted to be zero. The researchers demonstrated that the transition from a static state to a dynamic, time-varying state is not smooth in the way one might expect; the mathematical functions describing the response behave differently at the exact moment of zero frequency compared to when the frequency is merely very small. This suggests that the way these objects react to tides is deeply tied to the specific geometry of the flow and the rotation of the system.

To verify their calculations, the researchers compared their analytical formulas with numerical simulations of the fluid flow. They confirmed that their mathematical predictions accurately described the behavior of the waves near the event horizon, capturing the rapid oscillations that occur as the fluid accelerates. The study also highlighted that these fluid models are not just theoretical exercises but are physically realizable. Experiments with water in a bathtub or with super-cold atomic gases known as Bose-Einstein condensates can create these rotating flows. In such experiments, scientists could potentially create a localized disturbance to act as a tidal force and measure the resulting deformation of the flow. This would allow for a direct observation of how an event horizon responds to external stresses, something that is currently impossible to do with astrophysical black holes.

The implications of these findings extend beyond the laboratory. By showing that analogue black holes possess non-zero tidal Love numbers and exhibit specific logarithmic behaviors, the study provides a new benchmark for testing theories of gravity. If future observations of gravitational waves from real black hole mergers reveal similar non-zero tidal responses, it could indicate that our understanding of gravity needs to be revised, or that black holes have a more complex internal structure than previously thought. The fluid models serve as a bridge, allowing physicists to explore these extreme conditions in a controlled environment. The research confirms that while the equations governing these fluid flows are different from those of gravity, the resulting phenomena offer a powerful way to visualize and test the fundamental properties of black holes and white holes, turning the abstract mathematics of the cosmos into something that can be studied in a tank of water.

Technical Summary: Dynamical Love numbers of analogue rotating black and white holes

Problem Statement
The paper addresses the calculation of dynamical tidal response coefficients (TRCs) for 2+1 dimensional analogue black holes (ABHs) and analogue white holes (AWHs). While static tidal Love numbers (TLNs) for asymptotically flat black holes in Einstein gravity are known to be identically zero, and static tidal dissipation coefficients (TDCs) can be non-zero due to frame-dragging, the behavior of these coefficients in the dynamical regime (time-dependent perturbations) for analogue spacetimes remains to be systematically explored. Furthermore, unlike general relativity, fluid dynamical analogue gravity permits the physical existence of white hole event horizons. The authors aim to compute the TRCs for both ABHs and AWHs generated by rotating draining and fountaining bathtub flows, specifically analyzing the transition from static to dynamical regimes and the relationship between the TRCs of the black and white hole counterparts.

Methodology
The authors employ a barotropic, irrotational, inviscid, constant-density fluid model in a 2+1 dimensional bathtub geometry.

Background Spacetime: They derive the effective acoustic metric for a time-independent, torque-free, axisymmetric flow. This setup yields a pair of solutions: a draining flow (ABH) and a fountaining flow (AWH), which share conserved quantities (angular momentum, Bernoulli constant) but possess opposite mass flow rates.
Perturbation Equation: Linear perturbations of the fluid are treated as massless Klein-Gordon scalar fields satisfying a wave equation in the effective acoustic metric. Using a Teukolsky-inspired separation of variables, the authors derive a radial wave equation (Eq. 5.4) dependent on frequency $\omega$ and azimuthal number $m$ .
Analytical Approximation: To solve the radial equation in the near-zone limit ( $\omega r \ll 1$ $ω r ≪ 1$ ), the authors expand the solution in powers of the frequency $\omega$ $ω$ :
- Zeroth Order ( $\omega^0$ ): The static limit is solved analytically using Gauss hypergeometric functions.
- First Order ( $\omega^1$ ): The dynamical correction is obtained by truncating the equation at linear order in $\omega$ . The authors impose ingoing boundary conditions for the ABH and outgoing boundary conditions for the AWH at the respective horizons.
Extraction of Coefficients: The TRCs are extracted by comparing the asymptotic behavior of the radial solutions at large distances with the standard definition of tidal potentials. The real part of the TRC is identified as the TLN ( $\kappa$ ), and the imaginary part as the TDC ( $\nu$ ).
Numerical Verification: The authors perform numerical integration of the full radial equation to validate the analytical approximations, particularly checking the agreement in the near-zone region and the behavior of the extracted coefficients.

Key Contributions and Results

Dynamical TRCs for ABHs: The authors derive explicit analytical expressions for the TRCs of rotating ABHs. They find that in the static limit ( $\omega \to 0$ ), the TLN exhibits logarithmic running with the radial coordinate ( $\ln r$ ), while the static TDC vanishes. This contrasts with Einstein gravity where static TLNs vanish entirely.
Dynamical TRCs for AWHs: A significant finding is the relationship between the TRCs of the ABH and AWH. For a given pair of solutions with the same rotation parameters, the TRC of the AWH is the complex conjugate of the TRC of the ABH. Consequently, the TLNs are identical ( $\kappa_{ABH} = \kappa_{AWH}$ ), while the TDCs are equal in magnitude but opposite in sign ( $\nu_{AWH} = -\nu_{ABH}$ ).
Static-Dynamic Transition: The paper highlights that the transition from static to dynamical regimes is non-trivial. The TRC as a function of frequency is not analytic at $\omega = 0$ . Specifically, as $\omega \to 0$ , the TLN diverges (due to the logarithmic term) while the TDC approaches a finite value (or zero depending on the specific limit path), whereas the underlying radial wave equation solution remains smooth, with $\omega=0$ acting as a removable singularity.
Parameter Dependence: The TRCs are shown to be complex-valued functions of the frequency $\omega$ , the azimuthal number $m$ , and the rotation parameter $l$ . The authors identify specific constraints on these parameters required for the validity of the linear $\omega$ approximation.
Numerical Confirmation: Numerical results confirm that the first-order analytical solution provides a better approximation to the full numerical solution than the zeroth-order (static) solution, particularly in capturing the near-horizon oscillations and the transition to the dynamical regime.

Significance and Claims
The paper claims to provide the first systematic calculation of dynamical TRCs for analogue black and white holes, extending the study of tidal effects beyond the static regime and beyond the black hole sector to include white holes. The authors emphasize that fluid dynamics allows for the existence of AWHs, enabling the study of time-reversed tidal responses.

The results demonstrate that unlike asymptotically flat black holes in Einstein gravity (where static TLNs are zero), analogue black holes exhibit non-zero static TLNs with logarithmic running. The paper suggests that these findings could be relevant for laboratory experiments using surface waves in draining bathtubs or Bose-Einstein condensates (BECs), where observers have access to all regions of the spacetime, including the interior and near-horizon zones. The authors note that the rapid oscillation of waves near the horizon (visible in their numerical plots) could potentially allow for the observation of trans-Planckian signatures in future experimental extractions of TRCs. The work establishes a theoretical framework for understanding how tidal deformability and dissipation manifest in analogue gravity systems, offering a potential pathway to test concepts related to black hole horizons in controlled laboratory settings.

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